Helicoid

The helicoid is also a ruled surface (and a right conoid), meaning that it is a trace of a line.

Indeed, Catalan proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces.

[1][2] A helicoid is also a translation surface in the sense of differential geometry.

The helicoid and the catenoid are parts of a family of helicoid-catenoid minimal surfaces.

The helicoid is shaped like Archimedes screw, but extends infinitely in all directions.

If α is positive, then the helicoid is right-handed as shown in the figure; if negative then left-handed.

A helicoid with α = 1, −1 ≤ ρ ≤ 1 and − π ≤ θ ≤ π .
Animation showing the local isometry of a helicoid segment and a catenoid segment.